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Catalan's conjecture

The Catalan conjecture states that apart from the powers 23 = 8 and 32 = 9 there are no other real powers that differ by exactly 1.

 


Explanation

The conjecture was formulated in 1844 by the Belgian mathematician Eugène Charles Catalan. The calculation can be written as

32 − 23 = 9 − 8 = 1

 


Additional information

In 2002 the Romanian mathematician Preda Mihăilescu proved that the only solution of two consecutive powers in the natural numbers for

xa − yb = 1

with x, y, a, b > 1 actually is x = 3, a = 2, y = 2 and b = 3. That is why it is now also called the theorem of Mihăilescu.

 


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